On the pciture is winged hussar - so he is definatelly not a light cavalry class.

In the old Commonwealth to fight steppe archers was used light cavalry which made about 20% of all cavalry forces.
On the begining there was two types of those cavalry in the Polish army: Light Tatar Cavalry and Light Wallahian Cavalry. Later was created also Light Cossack cavalry.
They were usually armed with lance, pistols, bow and sabre,
shield.
In the second half of XVIII century all those types of Polish light cavalry (but especially Tatar) changed into uhlans which became popular type of cavalry in the whole Europe and even overseas (the word ulhlan also comes from Tatar language).

The Han Dynasty approach was to raise their own horses. Other than matching them with horses, no.

Here the basic reasons:

1. Horses are too fast for foot troops. Even if a nomadic army is losing, they can also safely escape against a foot army.
2. Archers deal damage at range. Only foot archers can really damage Horse armies.
3. Horse army can easily wipe out a routed infantry army.

The only battles in which horse archers had crucial losses was the result of a poor commander or hostile terrain.

Generally so, but massing ancient armies was a serious business, most ancient battles happen because both sides want to resolve things now, or they are forced into the field (defending something whatever), Gathering a large Army is costly, (even for Horse nomads) and refusing battle, withdrawing is admitting defeat with the consequent re-evaluation of real strength by everyone involved (allies and subjects are much less inclined to show up next time) IF the army includes anything that isnt cavalry it's abandoned and lost.

Battles not fought in large open spaces present problems, a plain with few large exists would mean through faster cavalry can not withdraw quickly, prime example when facing an opponent on and exit valley from a Mountain range, common enough occurrence occasionally one must cross mountains to fight the enemy, best to do it away from the enemy force and approach over less dense terrain. But fast Marching of armies is often about organisation and a foot army might well be better organised and appear sooner than one thought as one exists the Mountains.

The Han Dynasty approach was to raise their own horses. Other than matching them with horses, no.

Here the basic reasons:

1. Horses are too fast for foot troops. Even if a nomadic army is losing, they can also safely escape against a foot army.
2. Archers deal damage at range. Only foot archers can really damage Horse armies.
3. Horse army can easily wipe out a routed infantry army.

The only battles in which horse archers had crucial losses was the result of a poor commander or hostile terrain.

And that only applies when the other army has caught the horsemen for a pitched battle. Which, if the cav have it their way, wouldn't happen most of the time.

agreed that horse armies are pretty hard to defeat if they simply choose not to engage you. Even if you break them, they can just slip away. Defending against them is one thing.

No, no, you assume that the only way of engaging is a pitche battle - which is simply not true. Cavalry armies are known to have defeated much larger forces in detail by harrassment and/or swift surprise action.